Definition:Coimage (Category Theory)

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Let $\CC$ be a locally small category.

Let $f : X \to Y$ be a morphism in $\CC$.

A coimage of $f$ consists of an object $C$ and an epimorphism $e: X \to C$ such that:

$(1): \quad$ There exists a morphism $m : C \to Y$, such that $f=m \circ e$.
$(2): \quad$ For any object $C'$ with a morphism $m' : C' \to Y$ and an epimorphism $e' : X \to C'$ such that $f = m' \circ e'$, there exists a unique morphism $v : C' \to C$ such that $e = v \circ e'$.